108 LECTURE XIII. 



longitudinal forces ; and there is no doubt but that some of them were 

 occasioned by the difficulty of applying the force precisely at the extremi- 

 ties of the axis, and others by the accidental inequalities of the substances, 

 of which the fibres must often have been in such directions as to constitute 

 originally rather bent than straight columns. 



When a rod, not very flexible, is fixed at one end in a horizontal posi- 

 tion, the curvature produced by its own weight is every where as the 

 square of the distance from the other end : and if a rod be simply sup- 

 ported at each end, its curvature at any point will be proportional to the 

 product of the two parts into which that point divides it. But when the 

 weights are supposed to be applied to any given points of the rod only, the 

 curvature always decreases uniformly between these points and the points 

 of support. (Plate IX. Fig. 122, 123.) 



The stiffness of any substance is measured by the force required to cause 

 it to recede through a given small space in the direction of the force. It is 

 only necessary to consider this property with regard to forces applied 

 transversely. In such cases the stiffness is directly as the breadth and the 

 cube of the depth of the beam, and inversely as the cube of its length.* 

 Thus if we have a beam which is twice as long as another, we must make 

 it, in order to obtain an equal stiffness, either twice as deep or eight times 

 as broad. The property of stiffness is fully as useful in many works of 

 art as the ultimate strength with which a body resists fracture : thus for a 

 shelf, a lintel, or a chimney piece, a great degree of flexure would be 

 almost as inconvenient as a rupture of the substance. 



When a beam is supported at both ends, its stiffness is twice as great as 

 that of a beam of half the length firmly fixed at one end ; and if both ends 

 are firmly fixed the stiffness is again quadrupled. For if the whole beam 

 were inverted and supported by a fulcrum in the middle, each half would 

 resemble a separate beam fixed at one end, and the fulcrum would bear 

 the sum of two equal weights placed at the extremities, disregarding that 

 of the beam ; and consequently the same flexure will be produced by 

 placing a double weight on the middle of the beam in an inverted position. 

 If both ends were firmly fixed, the curvature would be every where as the 

 distance from the middle of each half, the whole being in the same state 

 as four separate beams fixed at their extremities : each of these beams 

 would be eight times as stiff as beams of twice the length, and the whole 

 beam, in this state, would be eight times as stiff as if the ends were simply 

 supported. It is, however, difficult to fix the ends of a beam so firmly as 

 to increase its resistance in this proportion, unless it be continued both 

 ways considerably beyond the supports. 



It is evident that a tube or hollow beam of any kind, must be much 

 stiffer than the same quantity of matter in a solid form : the stiffness is 

 indeed increased nearly in proportion to the square of the diameter, since 

 the cohesion and repulsion are equally exerted with a smaller curvature, 

 and act also on a longer lever. 



Torsion, or twisting, consists in the lateral displacement or detrusion of 

 the opposite parts of a solid, in opposite directions, the central particles 

 * Robison's Mechanical Philosophy, art. Strength of Materials, 386. 



